A and B together can complete a certain work in 54 days. They work together for 12 days. The remaining work was completed by A, B and C together in 28 days. If the work done by B in 6 days is equal to the work done by C in 9 days, then A alone can complete two-thirds of the work in ___________ days.

Correct option is D

Given:

• A and B together can complete the work in 54 days
• A and B work together for 12 days
• Remaining work is completed by A, B, and C in 28 days
• Work done by B in 6 days = C in 9 days
• We need to find how many days A alone will take to complete two-thirds of the work

Formula Used:

1. Work = Rate × Time
2. If person X completes work in D days, then
   X's 1-day work = 1/D
3. To find remaining work:
   Remaining Work = 1 − Work already done
4. Proportionality from “equal work” condition:
   If B’s 6-day work = C’s 9-day work → 6B = 9C → relate efficiencies

Solution:

$$\text{A + B's 1-day work} = \frac{1}{54} \\[1.5ex]\text{Work done in 12 days} = 12 \times \frac{1}{54} = \frac{2}{9} \\[1.5ex]\text{Remaining work} = 1 - \frac{2}{9} = \frac{7}{9} \\[1.5ex]\text{A + B + C complete } \frac{7}{9} \text{ of the work in 28 days} \\\Rightarrow \text{Their 1-day work} = \frac{7}{9} \div 28 = \frac{1}{36} \\[1.5ex]\text{Given: B's 6-day work = C's 9-day work} \\\Rightarrow 6B = 9C \Rightarrow \frac{B}{C} = \frac{3}{2} \Rightarrow C = \frac{2}{3}B \\[1.5ex]\text{Now,} \quad A + B = \frac{1}{54} \\A + B + C = A + B + \frac{2}{3}B = A + \frac{5}{3}B = \frac{1}{36} \\[1.5ex]\text{Subtracting the two equations:} \\\left(A + \frac{5}{3}B\right) - \left(A + B\right) = \frac{1}{36} - \frac{1}{54} \\\ \\\Rightarrow \frac{2}{3}B = \frac{1}{108} \Rightarrow B = \frac{1}{72} \\[1.5ex]\text{Substitute into } A + B = \frac{1}{54}: \\\ \\A + \frac{1}{72} = \frac{1}{54} \Rightarrow A = \frac{1}{54} - \frac{1}{72} \\\ \\= \frac{4 - 3}{216} = \frac{1}{216} \\[1.5ex]\text{So, A's 1-day work} = \frac{1}{216} \\[1.5ex]\text{Time to complete } \frac{2}{3} \text{ of the work} = \frac{2}{3} \div \frac{1}{216} = \frac{2}{3} \times 216 = \boxed{144 \text{ days}}$$

​Final Answer: (D) 144 days